Transversely isotropic model for wellbore stability analysis in laminated formations

ABSTRACT

A method of predicting wellbore stability is provided and includes: creating a parameterized model of a wellbore in laminated formation, the parameterized model including a plurality of laminated formation and wellbore related parameters; considering measurement data to determine the laminated formation and wellbore related parameters; updating the parameterized model by adopting the determined laminated formation and wellbore related parameters; and applying the updated parameterized model to derive a solution of wellbore stability.

FIELD OF THE INVENTION

The present invention relates generally to oil and gas well drilling andhydrocarbon production. More particularly, the present invention relatesto a model of predicting wellbore stability in laminated formations.

BACKGROUND OF THE INVENTION

Stress-induced wellbore failures are common in oilfield exploration andproduction. This problem has been an important concern for operators, asborehole collapses and lost circulation during well drilling can causeeconomic losses. Such stress-induced wellbore failures can also causeproblems in production operations, as sanding control and management areof utmost importance for the economy of a field. Therefore, to safelydrill a deep well for hydrocarbon exploration or production, it isnecessary to predict the wellbore stability and avoid wellbore failures,and a better understanding of the rock mechanical properties and failurebehavior is essential to improve the economics of an oilfielddevelopment.

Most of current wellbore stability models are based on the assumptionthat formation rock is a continuous isotropic medium, thus thetraditional shear/tensile failure models of intact rockmass are used.For example, Bradley (1979) laid a milestone for inclined wellboreanalysis by providing an analytical solution based on linear, isotropicelasticity.

However, in most cases anisotropy behavior of formation is found, andwellbore failures related to bedding or laminated formations have beenwidely recognized as a common cause of wellbore instability. Thus it isdifficult to apply the theory based on the assumption that formationrock is a continuous isotropic medium on the actual analysis oflaminated formations.

To approach the anisotropy behavior of formation, the mechanics ofanisotropic material was introduced into engineering in early the 20thcentury, and much research has been carried out in the area of findingthe proper Green's functions to describe the elastic displacementresponse of a linear elastic medium to the applied force. For example,for transversely isotropic materials, the 3-D Green's functions in afull-space have been obtained. In addition, a generalized formalism toexpress the deformation of dislocations and cracks in an anisotropicmedium has also been obtained. Unfortunately, due to the mathematicaldifficulty, there is no unique analytical solution to the problem of aborehole embedded in a transversely isotropic material.

Further, several models of the deformation and failure oflaminated/bedding rock around a cylindrical cavity have been developed.For example, Aadnoy treats rock as a transversely isotropic elasticmaterial with failure been governed by a Mohr-Coulomb criterionincorporating a single plane of weakness. However, Aadnoy's model hasdisadvantage that it simplified the transversely isotropic model byusing only three elastic parameters, rather than the five that areactually required. Willson et al. also adopted the single plane failuremodel in analyzing the wellbore stability problems in beddingformations, but unrealistically assumed isotropic elastic behavior whencomputing the deformation and induced stresses in their model.

Still another approach is to use the existing numerical methods, such asthe 3-D Finite Element Method (FEM), to solve for the stress anddeformation around a borehole embedded in a laminated formation, andthen predict the borehole stability by appropriately incorporating thefailure criterion. However, these numerical methods have not beenapplied to field applications systemically. There are two reasons.First, in laminated formations, boundaries or interfaces betweendifferent formations pose great difficulties in discretization of theelements, especially in the cases when the borehole axis is inclined tothe axis of the laminated formation plane. Second, 3-D FEM is acomputational time costly approach and it is impossible to use thismethod in a log-based analysis.

SUMMARY OF THE INVENTION

It is therefore an object of the invention to provide methods andapparatus for the prediction of wellbore stability in laminatedformations.

It is another object of the invention to provide methods and apparatusfor the prediction of wellbore stability in laminated formations whichtakes into account the laminated formation and wellbore relatedparameters of the wellbore and the field.

It is a further object of the invention to provide methods and apparatusfor the prediction of wellbore stability in laminated formations whichcan be applied in a log-based analysis.

It is still another object of the invention to develop a semi-analyticalmathematical model to compute the deformation and induced stressesaround a borehole in transversely isotropic formations.

It is an additional object of the invention to provide methods andapparatus for proposing a hybrid failure criterion for the wellborestability.

It is still another object of the invention to propose a failure modelwhich takes the failure mechanisms of laminated formations into accountto predict borehole instability or sanding.

In accord with the objects of the invention which will be discussed inmore detail below, a method of predicting wellbore stability is providedand includes: creating a parameterized model of a wellbore in laminatedformation, the parameterized model including a plurality of laminatedformation and wellbore related parameters; considering measurement datato determine the laminated formation and wellbore related parameters;updating the parameterized model by adopting the determined laminatedformation and wellbore related parameters; and applying the updatedparameterized model to derive a solution of wellbore stability.

In a preferred embodiment of the invention, the method further includesrepeating the considering and updating steps with additional measurementdata to produce a further updated parameterized model to derive afurther solution of wellbore stability.

Additional objects and advantages of the invention will become apparentto those skilled in the art upon reference to the detailed descriptiontaken in conjunction with the provided figures.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is illustrated by way of example and not intendedto be limited by the figures of the accompanying drawings in which likereferences indicate similar elements and in which:

FIG. 1 is a flowchart showing steps associated with the present method,apparatus, and article of manufacture;

FIG. 2 is a schematic illustration of computer hardware associated withthe apparatus and article of manufacture;

FIG. 3 is a diagram used to illustrate the material coordinate system oftransversely isotropic material;

FIG. 4 is a diagram used to illustrate the borehole orientation andcoordinate system;

FIG. 5 is a diagram used to illustrate the laminated formation strikeand dip;

FIG. 6 is a diagram used to illustrate the intact rock shear failure andweakness plane slippage shear failure;

FIG. 7 is a diagram used to illustrate an example of the result of thepresent method to indicate induced stresses around wellborecircumference;

FIG. 8 is a group of diagrams used to illustrate another example of theresult of the present method to indicate the displacements of wellborecircumference; and

FIG. 9 is a group of diagrams used to illustrate another example of theresult of the present method to indicate wellbore stabilities versusdeviation.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 shows several steps associated with the present method, apparatusand article of manufacture and provides a general overview of theinvention. In the Create Parameterized Model Step 10, a parameterizedmodel of the wellbore in laminated formation is created. Theparameterized model includes a plurality of laminated formation andwellbore related parameters. In one embodiment of the invention, thelaminated formation and wellbore related parameters include 5 elasticparameters of laminated formation, e.g. E, E′, ν, ν′ and G′.Respectively, E and E′ are the Young's modulus with respect todirections lying in the plane of isotropy and perpendicular to it; ν isthe Poisson coefficient characterizing the transverse reduction in theplane of isotropy for tension in the same plane; ν′ is the Poissoncoefficient characterizing the transverse reduction in the plane ofisotropy for tension in a direction normal to it; and G′ is the shearmodulus for planes normal to the plane of isotropy. The details of thetheory and how the model is created will be explained by followingdrawings and descriptions.

In the Consider Measurement Data to Determine Parameters Step 14,Measurement Data 12 that provides information regarding the formationand wellbore is examined. In one embodiment, Measurement Data 12 isconsidered to determine the laminated formation and wellbore relatedparameters. Again, the details of the theory and how the model iscreated will be explained by following drawings and descriptions.

In the Predict Wellbore Stability Step 16, the parameterized model isupdated using the determined laminated formation and wellbore relatedparameters. The parameterized model is then used to derive the solutionof wellbore stability. In another embodiment, data of field in-situstress derived from a geomechanical model is applied to theparameterized model to derive the solution of wellbore stability.

The Consider Measurement Data to Determine Parameters Step 14 may berepeated as desired using Additional Measurement Data 18 to furtherupdate the laminated formation and wellbore related parameters and theparameterized model, as shown in Consider Measurement Data to UpdateParameters & Model Step 14. This may be repeated, for instance, whenevera new set of Additional Measurement Data 18 from the subsurface areabecomes available. The further updated parameterized model is used toderive a further solution of wellbore stability, as shown in PredictWellbore Stability Step 22.

The Measurement Data 12 and the Additional Measurement Data 18 may, forinstance, consist of different types of data, such as seismic data,drilling data, well logging data, well test data, production historydata, permanent monitoring data, or the field boundary conditions of theformation and wellbore. The Measurement Data 12 and the AdditionalMeasurement Data 18 may, alternatively, consist of the same type of datathat has been acquired from the subsurface area at different times tomeasure changes in reservoir.

The above method can be applied in a log-based analysis to derive thesolution of wellbore stability as well as in real time drilling. It canalso propose a hybrid failure criterion for the wellbore stability.

FIG. 2 schematically illustrates computer hardware that may be used toimplement the inventive method. Computer 30 has a media reading device,such as a CD-ROM Reader 32, a floppy disk device, or a ZIP drive. Themedia reading device may also be capable of recording the output of theprogram the computer 30 is running. A user of the computer 30 may entercommands using a user input device, such as a keyboard 34 or a mouse,may view output of the program code on a visual display device, such asmonitor 36, and may make hardcopies of output using an output device,such as printer 38. When properly configured, computer 30 (and itsassociated peripheral devices) is an apparatus for creating a model of asubsurface area in accordance with the present invention. Computermedia, such as a CD-ROM 40, a floppy disk, or a ZIP disk, may havecomputer readable program code that allows the computer 30 to create amodel of a subsurface area in accordance with the inventive method.

A preferred embodiment of the inventive method will now be described insubstantially more detail. The inventive method addresses two primaryissues: how to create a parameterized model of the laminated formationand wellbore, and to propose a failure model which takes the failuremechanisms of laminated formations into account to predict boreholeinstability or sanding.

The first example will be used to illustrate how to create aparameterized model of the laminated formation and wellbore. Theparameterized model of the laminated formation and wellbore is asemi-analytical mathematical model which idealizes the problem byconsidering a cylindrically cavity embedded in a homogeneoustransversely elastic medium under the effects of in-situ stresses andborehole pressure. Based on the model developed by Stroh, a complexfunction of displacements that satisfies the elastic constitutions oftransverse isotropy medium, and force equilibrium conditions, isdeveloped to solve the induced stresses around the borehole. This systemand its algorithm can be used in the log analysis to simulate wellborestability conditions and predict the onset of sanding. In this system,borehole trajectory, formation laminated planes and the directions ofin-situ stresses all can be arbitrarily orientated to each other tomodel the actual field situation. This model includes independentlaminated formation related parameters: E, E′, ν, ν′, and G′.

The generalized form of the anisotropic linear elasticity constitutioncan be expressed as:σ_(ij) =c _(ijkl)ε_(kl) =c _(ijkl) ∂u _(k) /∂x _(l) (i, j, k, l=1, 2, 3)   (1)where σ_(ij) and ε_(kl) are the stress and strain tensor respectively,C_(ijkl) is the elastic stiffness and u_(k) represent the displacements.

It should be noted that we follow the notation rule that negative normalstress (σ_(ii)<0) represents compression stress and positive normalstress represents tension stress.

FIG. 3 is a diagram illustrating the material coordinate system oftransversely isotropic material. If we assume the laminated formationrock to be continuous, homogeneous and transversely isotropic, theconstitutive relations can be expressed in the material coordinatesystem X₁X₂X₃ as: $\begin{matrix}{\begin{Bmatrix}ɛ_{11} \\ɛ_{22} \\ɛ_{33} \\{2ɛ_{23}} \\{2ɛ_{31}} \\{2ɛ_{12}}\end{Bmatrix} = {\begin{bmatrix}a_{11} & a_{12} & a_{13} & 0 & 0 & 0 \\a_{12} & a_{11} & a_{13} & 0 & 0 & 0 \\a_{13} & a_{13} & a_{33} & 0 & 0 & 0 \\0 & 0 & 0 & a_{44} & 0 & 0 \\0 & 0 & 0 & 0 & a_{44} & 0 \\0 & 0 & 0 & 0 & 0 & a_{66}\end{bmatrix}\begin{Bmatrix}\sigma_{11} \\\sigma_{22} \\\sigma_{33} \\\sigma_{23} \\\sigma_{31} \\\sigma_{12}\end{Bmatrix}}} & (2)\end{matrix}$where a_(ij) (ij=1,2, . . . 6) are components of the transverselyisotropic elastic compliance tensor.

The non-zero components are:a ₁₁=1/E; a ₁₂ =−ν/E; a ₁₃ =−ν′/E′;   (3)a ₃₃=1/E′; a ₄₄=1/G′; a ₆₆=2(1+ν)/Ewhere E, E′, ν, ν′ and G′ are the five independent elastic parameters.Respectively, E and E′ are the Young's moduli with respect to directionslying in the plane of isotropy and perpendicular to it; ν is the Poissoncoefficient characterizing the transverse reduction in the plane ofisotropy for tension in the same plane; ν′ is the Poisson coefficientcharacterizing the transverse reduction in the plane of isotropy fortension in a direction normal to it; and G′ is the shear modulus forplanes normal to the plane of isotropy, as shown in FIG. 3. Thus, allfive independent laminated formation related parameters (E, E′, ν, ν′,and G′) are considered.

FIG. 4 is a diagram illustrating the borehole orientation and coordinatesystem. Considering an orientated wellbore (with inclination w_(d) andazimuth w_(a), azimuth is referenced to North) that is embedded in atransversely isotropic medium, we can define a borehole coordinatesystem X′₁X′₂X′₃, as shown in FIG. 4, in which X′₃ is along the boreholeaxis and pointing upwards, and X′₁ pointing to the lowest part ofborehole wall in the vertical plane. The isotropic plane is defined by astrike angle b_(s) and a dip angle b_(d), as shown in FIG. 5, which candefine the material coordinate system X₁X₂X₃.

The geometric transformation tensor B, from the material coordinatesystem X₁X₂X₃ to borehole coordinate system X′₁X′₂X′₃ can be expressedas: $\begin{matrix}{\lbrack B\rbrack = \begin{bmatrix}B_{11} & B_{12} & B_{13} \\B_{21} & B_{22} & B_{23} \\B_{31} & B_{32} & B_{33}\end{bmatrix}} & (4)\end{matrix}$

Within the wellbore coordinates X′₁X′₂X′₃, the elastic compliance tensora′_(ij) can be obtained by: $\begin{matrix}{a_{i\quad j}^{\prime} = {\sum\limits_{m = 1}^{6}{\sum\limits_{n = 1}^{6}{a_{m\quad n}q_{m\quad i}q_{n\quad j}}}}} & (5)\end{matrix}$where q_(ij) is a geometric transformation function and can be obtainedfrom the geometric transformation tensor B as: $\begin{matrix}{q_{i\quad j} = \begin{pmatrix}B_{11}^{2} & B_{21}^{2} & B_{31}^{2} & {2B_{21}B_{31}} & {2B_{31}B_{11}} & {2B_{11}B_{21}} \\B_{12}^{2} & B_{22}^{2} & B_{32}^{2} & {2B_{22}B_{32}} & {2B_{32}B_{12}} & {2B_{12}B_{22}} \\B_{13}^{2} & B_{23}^{2} & B_{33}^{2} & {2B_{23}B_{33}} & {2B_{33}B_{13}} & {2B_{13}B_{23}} \\{B_{12}B_{22}} & {B_{22}B_{23}} & {B_{32}B_{33}} & {{B_{22}B_{33}} +} & {{B_{12}B_{33}} +} & {{B_{12}B_{23}} +} \\\quad & \quad & \quad & {B_{32}B_{23}} & {B_{32}B_{13}} & {B_{22}B_{13}} \\{B_{13}B_{11}} & {B_{23}B_{21}} & {B_{33}B_{31}} & {{B_{23}B_{31}} +} & {{B_{13}B_{31}} +} & {{B_{13}B_{21}} +} \\\quad & \quad & \quad & {B_{33}B_{21}} & {B_{33}B_{11}} & {B_{23}B_{11}} \\{B_{11}B_{12}} & {B_{21}B_{22}} & {B_{31}B_{32}} & {{B_{21}B_{32}} +} & {{B_{11}B_{32}} +} & {{B_{11}B_{22}} +} \\\quad & \quad & \quad & {B_{31}B_{22}} & {B_{31}B_{12}} & {B_{21}B_{12}}\end{pmatrix}} & (6)\end{matrix}$

The flowing equations are expressed in the borehole coordinate systemX′₁X′₂X′₃. For the simplification of the derivation, we will name thesystem as x1x2x3 later on.

Assuming the generalized plane strain condition in the planeperpendicular to borehole axis, and following Stroh's formalism, thedisplacements of rock around the borehole have the form:u _(k) =A _(k) f(x ₁ +Px ₂) (k=1,2,3)   (7)where ƒ is an analytical function of the complex variable Z=x₁+ix₂, Pand A_(k) are coefficients.

The equilibrium equations ∂σ_(ij)/∂x_(i)=0 require:(C _(i1k1) +P(C _(i1k2) +C _(i2k1))+P ² C _(i2k2))A _(k)=0 (i, k=1,2,3)  (8)

Non-trivial values of A_(k) occur if P satisfies the equation:|C _(i1k1) +P(C _(i1k2) +C _(i2k1))+P ² C _(i2k2)|=0   (9)

Compacting the C_(ijkl) tensor following Voigt's recipe, the aboveequations can be rewritten as:(P ² g ₁₁−2P ³ g ₁₆ +P ²(2g ₁₂ +g ₆₆)−2Pg ₂₆ +g ₂₂)(P ² g ₅₅−2Pg ₄₅ +g₄₄)−(P ³ g ₁₅ −P ²(g ₁₄ +g ₅₆)+P(g ₂₅ +g ₄₆)−g ₂₄)²=0   (10)whereg _(ij) =a′ _(ij) −a′ _(i3) a′ _(j3) /a′ ₃₃ (i,j=1,2,3,4,5,6)   (11)

The roots of the equation (10) occur in conjugate pairs. The three rootswith positive imaginary parts will be denoted by P_(α) (α=1,2,3), withcorresponding eigenvectors been denoted by A_(kα).

Variables Z_(α) are defined as:Z _(α) =x ₁ +P _(α) x ₂   (12)

Now we can express the displacements as: $\begin{matrix}{u_{k} = {{\sum\limits_{\alpha}{A_{k\quad\alpha}{f_{\alpha}\left( z_{\alpha} \right)}}} + {\sum\limits_{\alpha}{{\overset{\_}{A}}_{k\quad\alpha}{\overset{\_}{f_{\alpha}}\left( \overset{\_}{z_{\alpha}} \right)}}}}} & (13)\end{matrix}$

The stresses are given from the generalized Hook's law: $\begin{matrix}{\sigma_{i\quad 1} = {{- {\sum\limits_{\alpha}{L_{i\quad\alpha}P_{\alpha}{f_{\alpha}^{\prime}\left( z_{\alpha} \right)}}}} - {\sum\limits_{\alpha}{{\overset{\_}{L}}_{i\quad\alpha}{\overset{\_}{P}}_{\alpha}{\overset{\_}{f_{\alpha}^{\prime}}\left( \overset{\_}{z_{\alpha}} \right)}}}}} & (14)\end{matrix}$ $\begin{matrix}{\sigma_{i\quad 2} = {{\sum\limits_{\alpha}{L_{i\quad\alpha}{f_{\alpha}^{\prime}\left( z_{\alpha} \right)}}} + {\sum\limits_{\alpha}{{\overset{\_}{L}}_{i\quad\alpha}{\overset{\_}{f_{\alpha}^{\prime}}\left( \overset{\_}{z_{\alpha}} \right)}}}}} & (15)\end{matrix}$

As σ₁₂=σ₂₁, it then implies:L _(1═) +P _(α) L _(2α)=0   (16)where L_(iα) are defined by: $\begin{matrix}{L_{i\quad\alpha} = \begin{pmatrix}{- P_{1}} & {- P_{2}} & {- P_{3}} \\1 & 1 & 1 \\l_{1} & l_{2} & l_{3}\end{pmatrix}} & (17)\end{matrix}$

The coefficients la are determined by: $\begin{matrix}{l_{\alpha} = \frac{{P_{\alpha}^{3}g_{15}} - {P_{\alpha}^{2}\left( {g_{14} + g_{66}} \right)} + {P_{\alpha}\left( {g_{26} + g_{46}} \right)} - g_{24}}{{P_{\alpha}^{2}g_{55}} + {2P_{\alpha}g_{45}} - g_{44}}} & (18)\end{matrix}$

In the case when one root of l_(α) solved from the above equation (18)equals infinity, i.e. one P_(α) satisfies:P _(α) ² g ₅₅+2P _(α) g ₄₅ −g ₄₄=0   (19)

Then let the P_(α) be donated as P₃, The L_(iα) are then given by:$\begin{matrix}{L_{i\quad\alpha} = \begin{pmatrix}{- P_{1}} & {- P_{2}} & 0 \\1 & 1 & 0 \\l_{1} & l_{2} & 1\end{pmatrix}} & (20)\end{matrix}$

Define complex variables γ_(α) as:γ_(α)=(i−P _(α))/(i+P _(α))   (21)

Then the complex variable Z_(α) can be rewritten as: $\begin{matrix}\begin{matrix}{z_{\alpha} = {x_{1} + {P_{\alpha}x_{2}}}} \\{= {\frac{\left( {1 - {{\mathbb{i}}\quad P_{\alpha}}} \right)}{2}\left( {z + {\gamma_{\alpha}\overset{\_}{z}}} \right)}}\end{matrix} & (22)\end{matrix}$

The above equations (13, 14, and 15) give the general form ofdisplacements and stresses for transversely isotropic material under theplane strain condition. With given elastic constants in wellborecoordinate system α′_(ij), coefficients P_(α) and L_(iα) can bedetermined.

Next let us analyze the wellbore problems. As the problem underconsideration comprises a wellbore (let us assume the radial length isa) bounded internally by |z|=a, it is convenient to generalize theapproach of Green and Zerna and define the complex variables ζ_(α)(α=1,2,3) such that: $\begin{matrix}{z_{\alpha} = {\frac{\left( {1 - {{\mathbb{i}}\quad P_{\alpha}}} \right)}{2}\left( {\zeta_{\alpha} + \frac{\gamma_{\alpha}a^{2}}{\zeta_{\alpha}}} \right)}} & (23)\end{matrix}$

The ζ_(α) planes are chosen so that the circles |ζ_(α)|=α correspond tothe circles of |z|=a in the z-plane, with |ζ_(α)|→∞ as |z|→∞. To everypoint in plane ζ_(α) on or outside the circle, there corresponds onepoint in the plane z (z=x₁+i x₂) on or outside this wellbore circle.

It can be seen from equation (23) that: $\begin{matrix}\begin{matrix}{{f^{\prime}\left( z_{\alpha} \right)} = {{f^{\prime}\left( \zeta_{\alpha} \right)}\frac{\mathbb{d}\zeta_{\alpha}}{\mathbb{d}z_{\alpha}}}} \\{= \frac{2{f^{\prime}\left( \zeta_{\alpha} \right)}}{\left( {1 - {{\mathbb{i}}\quad P_{\alpha}}} \right)\left( {1 - {\gamma_{\alpha}{a^{2}/\zeta_{\alpha}^{2}}}} \right)}}\end{matrix} & (24)\end{matrix}$

In plane polar coordinates z(r,θ), with x₁=r cos θ, x₂=r sin θ, thedisplacements and stresses can be written as:u _(r) +iu _(θ) =e ^(−iθ)(u ₁ +iu ₂)   (25)σ_(rr)+σ_(θθ)=σ₁₁+σ₂₂   (26)σ_(rr)−σ_(θθ)+2iσ _(rθ) =e ^(−2iθ)(σ₁₁−σ₂₂+2iσ ₁₂)   (27)

Thus combining the above equations with equations (14) and (15), yields:$\begin{matrix}{{\sigma_{\theta\theta} - {\mathbb{i}\sigma}_{r\quad\theta}} = {{\sum\limits_{\alpha}{\left( {1 + {{\mathbb{i}}\quad P_{\alpha}}} \right)\left( {1 - {{\mathbb{e}}^{{- 2}{\mathbb{i}\theta}}\gamma_{\alpha}}} \right)\frac{L_{2\alpha}{f_{\alpha}^{\prime}\left( \zeta_{\alpha} \right)}}{1 - {\gamma_{\alpha}{a^{2}/\zeta_{\alpha}^{2}}}}}} + {\sum\limits_{\alpha}{\left( {1 - {{\mathbb{i}}\quad{\overset{\_}{P}}_{\alpha}}} \right)\left( {1 + {{\mathbb{e}}^{{- 2}{\mathbb{i}\theta}}/{\overset{\_}{\gamma}}_{\alpha}}} \right)\frac{{\overset{\_}{L}}_{2\alpha}{{\overset{\_}{f}}_{\alpha}^{\prime}\left( {\overset{\_}{\zeta}}_{\alpha} \right)}}{1 - {{\overset{\_}{\gamma}}_{\alpha}{a^{2}/{\overset{\_}{\zeta}}_{\alpha}^{2}}}}}}}} & (28) \\{{\sigma_{r\quad r} + {\mathbb{i}\sigma}_{r\quad\theta}} = {{\sum\limits_{\alpha}{\left( {1 + {{\mathbb{i}}\quad P_{\alpha}}} \right)\left( {1 - {{\mathbb{e}}^{{- 2}{\mathbb{i}\theta}}\gamma_{\alpha}}} \right)\frac{L_{2\alpha}{f_{\alpha}^{\prime}\left( \zeta_{\alpha} \right)}}{1 - {\gamma_{\alpha}{a^{2}/\zeta_{\alpha}^{2}}}}}} + {\sum\limits_{\alpha}{\left( {1 - {{\mathbb{i}}\quad{\overset{\_}{P}}_{\alpha}}} \right)\left( {1 + {{\mathbb{e}}^{{- 2}{\mathbb{i}\theta}}/{\overset{\_}{\gamma}}_{\alpha}}} \right)\frac{{\overset{\_}{L}}_{2\alpha}{{\overset{\_}{f}}_{\alpha}^{\prime}\left( {\overset{\_}{\zeta}}_{\alpha} \right)}}{1 - {{\overset{\_}{\gamma}}_{\alpha}{a^{2}/{\overset{\_}{\zeta}}_{\alpha}^{2}}}}}}}} & (29)\end{matrix}$

The axial stress, σ_(zz), is derived from the plane strain constraintε_(zz)=0. Thus:σ_(zz)=σ_(zz0)−(a′ ₃₁σ_(rr) +a′ ₃₂σ_(θθ) +a′ ₃₄σ_(θz) +a′ ₃₅σ_(rz) +a′₃₆σ_(rθ)/) a′ ₃₃   (30)where σ_(zzθ) is the in-situ normal stress in Z direction.

Here we proposed an appropriate form for θ′(z_(a)) as: $\begin{matrix}\begin{matrix}{{f^{\prime}\left( z_{\alpha} \right)} = {C_{\alpha} + {\frac{a^{2}}{\zeta_{\alpha}^{2}}D_{\alpha}}}} & \left( {{\alpha = 1},2,3} \right)\end{matrix} & (31)\end{matrix}$where C_(α) and D_(α) are complex constants that are to be determinedfrom the boundary conditions.

The boundary condition at the wellbore wall, |z|=a, isσ_(rr)=P_(w)   (32)σ_(rθ)=0   (33)σ_(rz)=0   (34)where P_(w) is the well pressure.

The boundary condition of stress at far-field is the in-situ stresses,that is:σ_(rr)+σ_(θθ=σ) ^(∞) ₁₁+σ^(∞) ₂₂   (35)σ_(rr)−σ_(θθ)+2iσ _(rθ) =e ^(−3iθ)(σ^(∞) ₁₁+σ^(∞) ₂₂+2iσ ^(∞) ₁₂)   (36)σ_(zr)=(e ^(iθ)(σ^(∞) ₃₁ −iσ ^(∞) ₃₂)+e ^(−iθ)(σ^(∞) ₃₁ +iσ ^(∞) ₃₂))/2  (37)where σ^(∞) ₁₁, σ^(∞) ₂₂, σ^(∞) ₁₂, σ^(∞) ₃₁ and σ^(∞) ₃₂ are thein-situ stresses.

Each of the equations 32-37 can be split into real and imaginary parts.Thus, there are twelve equations for twelve unknowns (the real andimaginary parts of each C_(α) and D_(α) counting as one unknown).However, the imaginary part of equation (35) is identically satisfied,so the system is under-determined. The additional equation comes fromthe requirement that a rigid body motion does not affect the stresscondition. Thus: $\begin{matrix}{{\sum\limits_{\alpha}{{Im}\left( {\left( {1 + {{\mathbb{i}}\quad P_{\alpha}}} \right)C_{\alpha}} \right)}} = 0} & (38)\end{matrix}$

Thus using the above conditions, we can solve the complex constantsC_(α) and D_(α), and then compute the displacements and stress aroundthe wellbore.

Therefore, by assuming the laminated formation rock as continuous,homogeneous, and a transversely isotropic linear elastic medium, wederived the semi-analytical mathematical model based on Stroh's methodand the generalized plane-stain concepts. Complex variables are definedto describe the deformation field in the wellbore coordinate system.Using the equilibrium equations and boundary conditions, thedisplacements can be solved and hence the induced stresses are obtained.

The second example will be used to illustrate the failure model whichtakes the failure mechanisms of laminated formations into account topredict borehole instability or sanding. Specifically, the considerationthat the laminated formation is comprised of intact rockmass and theinterfaces between the intact rockmass, two types of shear failures,shear failure of intact rockmass and shear slippage along a singleinterface (or a weakness plane), are included in this proposed failuremodel, as shown in FIG. 6.

For intact rockmass shear failure, we assume that it is governed by theclassical Morh-Coulomb criterion, that is:−(σ₁ +P _(p))>=UCS−(σ₃ +P _(p))tan²(φ/2+π/4)   (39)where σ₁ and σ₃ are the maximum and minimum principal stressrespectively, P_(p) is the formation pore pressure, and UCS and φ arerockmass unconfined compressional strength and friction anglerespectively.

Shear slippage along the weakness plane is governed by the shearstrength of the plane. Let τ and σ_(n) donate the shear and normalstress applying on this single plane, thus the slippage failure willoccur when:|τ|>=Co−(σ_(n) +P _(p))tan Φ  (40)where C_(o) and Φ are the cohesive strength and the friction angle ofthe weakness plane respectively.

Wellbore shear failure can occur either by exceeding the strength of theintact rockmass or by exceeding the strength of the weakness plane. Inthis approach, we assume that the most risky failure mode will dominatethe failure mode of the formation rock. Once this failure occurs at aparticular position, it will propagate in some way and the stress statenear this point will be changed, and thus another failure mode will beshielded.

Here we define two dimensionless shear failure factors, ƒ_(intact) forintact rock and ƒ_(plane) for the weakness plane, to represent the riskof these two shear failures respectively: $\begin{matrix}{f_{intact} = \frac{\left( {{- \sigma_{1}} + \sigma_{3}} \right)}{{{ucs}\left( {1 - {\sin\quad\phi}} \right)} + {\left( {{- \sigma_{1}} - \sigma_{3} - {2P_{p}}} \right)\sin\quad\phi}}} & (41) \\{f_{plane} = \frac{\tau }{c_{o} - {\left( {\sigma_{n} + P_{p}} \right)\tan\quad\Phi}}} & (42)\end{matrix}$

As we can see from the above definitions, ƒ_(intact) and ƒ_(plane) arethe ratio of shear stress that tending to cause failure, over theresistance to failure. ƒ_(intact)=1 and ƒ_(plane)=1 represent thecritical conditions for intact rock shear failure and plane slippageshear failure respectively. The greater the value of shear failurefactors ƒ_(intact) and ƒ_(plane), the higher the risk that failure willoccur.

We now provide some experimental results of applying the parameterizedmodel of the laminated formation and the failure model to predictborehole instability or sanding. A few of case studies are provided,which yields further insights into this distinct mechanical behavior ofwellbore stability in laminated formations.

A typical set of test data was used in the following example using thecurrent system. In order to gain some basic understanding of theinfluences of Transversely Isotropy on wellbore stability, we willpresent the results of this numerical analysis systematically andcovering the listed areas:

Induced stresses around a wellbore;

Deformation characteristics of a wellbore, and

Wellbore trajectory sensitivity analysis

The mechanical parameters of the laminated formation and other in-situproperties are:

TI formation elastic constants:

E=9.4 GPa, E′=5.2 GPa, ν=0.21, ν′=0.34, G′=5.83 GPa

TI formation rock strength (UCS and friction angle):

UCS=18.5 MPa, φ=20

Weakness plane strength (Cohesion and friction angle):

Co=6.35 MPa, Φ=15

Laminated plane direction:

Strike b_(s)=0, Dip b_(d)=0

In-situ stresses:

Sv=69 MPa, SH=64.1 MPa,Sh=61.6 MPa, Azim of Sh=0, Pp=20 MPa

Induced Stress Around Wellbore

Referring to FIG. 7, the induced tangential and axial stresses aroundthe wellbore circumference versus azimuth (red lines) are displayed. Forcomparison reason, the results for the isotropic medium (using theproperties in symmetric plane) are also displayed in the same figure(blue lines). Specifically, red line with circle shows the tangentialstress for TI formation; red line with cross shows the axial stress forTI formation; blue line with circle shows the tangential stress forisotropic formation; and blue line with cross shows the axial stress forisotropic formation. The wellbore is in the symmetric plane (horizontalin this case) and aligned in the direction of minimum horizontal stress(Sh), and the well pressure is 28 MPa.

We can see that the anisotropy has a big influence on induced stressesin the cases where the wellbore is aligned in or near the direction ofsymmetric plane. Comparing with the isotropic formations, in theTransverse formations, the induced stresses around wellborecircumference are:

-   -   Stresses are more concentrated and the variations of        concentrated stresses are larger;    -   There are four points where tangential stress is compressional        concentrated, and four points where tangential stress is        extensional concentrated;    -   The positions where stresses are concentrated is not aligned        with stress directions        Deformation of the Wellbore

FIG. 8 displays the displacements of wellbore circumference, where thearray lines represent the directions and relative magnitudes ofdisplacement. The condition is same as described above. For comparisonpurpose, we shows the displacements of wellbore circumference oftransversely isotropic formation (part A) and isotropic formation (partB). We can find that due to the anisotropic properties of the formation,the wellbore is deformed trending squared shape. This behavior isdistinctly different from the isotropic case, when the borehole isdeformed elliptically and, the direction of ellipse is controlled byfar-field stress directions.

Trajectory Sensitivity Analysis

FIG. 9 displays the stability conditions of wellbore versus boreholedeviation. All the input data of the formation properties and in-situstresses are same, and the well pressure is 28 MPa. In FIG. 9, Parts Ato G respectively shows the results of stability at wellbore deviationof 0, 15, 30, 45, 60, 75 and 90 degree, where the color changing fromorange to dark red represents the changing of relative stableness of theformation from most stable to most unstable, and the areas inside theblack lines are shear failure zones.

From FIG. 9 we can conclude that, the worst wellbore trajectory,regarding to the stability, is the one that inclined to formationsymmetric plane with an angle of around 45 degree. When the wellboredeviation is around 45 degree, shear failure zones of the formation isbecoming wider and deeper, which means big instability risk.

In summary, from this numerical example using the proposed model, we candraw the following understanding of wellbore stability in laminatedformations:

-   -   The worst borehole direction is when the angle between the        borehole axis and the bedding planes is around 45 degree;    -   Breakouts cannot give the stress direction in an obvious manner        as in the case of the isotropic formations;    -   In laminated formation, shear failure or wellbore collapse may        occur and propagate deep into the formations;    -   The safe mud window (from the pressure that supports the        wellbore from collapsing to the pressure that will fracture the        formation) in transversely Isotropic formation is usually narrow        in comparison with that in isotropic formations

It was shown that the degree of anisotropy has certain influences on theinduced stress in the formation rock around the wellbore, but moreimportantly, it is the angle subtended between the wellbore axis andlaminated plane that strongly influences the wellbore stability. Theresult also shows that in laminated formations, the minimum horizontalstress is not perfectly aligned with the direction of breakout (shearfailure zone), and the shape and size of the breakout zone around aborehole is more severe than that in isotropic formations.

The foregoing description of the preferred and alternate embodiments ofthe present invention has been presented for purposes of illustrationand description. It is not intended to be exhaustive or limit theinvention to the precise examples described. Many modifications andvariations will be apparent to those skilled in the art. The embodimentswere chosen and described in order to best explain the principles of theinvention and its practical application, thereby enabling others skilledin the art to understand the invention for various embodiments and withvarious modifications as are suited to the particular use contemplated.It is intended that the scope of the invention be defined by theaccompanying claims and their equivalents. Symbol Definition σ_(ij)stress tensor ε_(kl) strain tensor C_(ijkl) elastic stiffness u_(k)displacements E and E′ Young's moduli υ and υ′ Poisson coefficient G′shear modulus b_(s) strike angle b_(d) a dip angle B geometrictransformation tensor a_(ij) & a′_(ij) elastic compliance tensors q_(ij)geometric transformation function ƒ analytical function P and A_(k)coefficients Z_(α) and ζ_(α) variables σ_(zz) axial stress ε_(zz) planestrain constraint C_(α) and D_(α) complex constants P_(w) well pressureσ^(∞) ₁₁, σ^(∞) ₂₂, σ^(∞) ₁₂, in-situ stresses σ^(∞) ₃₁ and σ^(∞) ₃₂ σ₁maximum compressional stress σ₃ minimum compressional stress P_(p)formation pore pressure UCS rockmass unconfined compressional strength φrockmass unconfined friction angle τ shear stress σ_(n) normal stressC_(o) cohesive strength Φ friction angle ƒ_(intact) intact rock shearfailure factor ƒ_(plane) weakness plane shear failure factors

1. A method of predicting wellbore stability comprising: a) creating aparameterized model of a wellbore in laminated formation, saidparameterized model including a plurality of laminated formation andwellbore related parameters; b) considering measurement data todetermine said laminated formation and wellbore related parameters; c)updating said parameterized model by adopting said determined laminatedformation and wellbore related parameters; and d) applying said updatedparameterized model to derive a solution of wellbore stability.
 2. Themethod of claim 1 further comprising repeating said considering andupdating steps with additional measurement data to produce a furtherupdated parameterized model to derive a further solution of wellborestability.
 3. The method of claim 1, wherein said laminated formationand wellbore related parameters include 5 elastic parameters oflaminated formation.
 4. The method of claim 1, wherein said laminatedformation and wellbore related parameters include Young's moduli alongwith and perpendicular to the plane of isotropy.
 5. The method of claim1, wherein said laminated formation and wellbore related parametersinclude Poisson coefficients along with and perpendicular to the planeof isotropy.
 6. The method of claim 1, wherein said laminated formationand wellbore related parameters include shear modulus perpendicular tothe plane of isotropy.
 7. The method of claim 1 further comprisingcomputing at least one constant of said parameterized model to derivethe solution of wellbore stability.
 8. The method of claim 1 furthercomprising proposing a hybrid failure criterion for said wellborestability.
 9. The method of claim 1, wherein said measurement datacomprise seismic data, drilling data, well logging data, well test data,production history data, or permanent monitoring data.
 10. The method ofclaim 1, wherein said measurement data comprise data of field boundaryconditions.
 11. The method of claim 1, wherein said measurement datacomprise data of field in-situ stress.
 12. The method of claim 2,wherein said measurement data and said additional measurement datacomprise different types of data.
 13. The method of claim 1, whereinsaid updating step comprises directly updating said parameterized modelusing additional measurement data obtained while drilling.
 14. Anapparatus for predicting wellbore stability comprising: a) means forcreating a parameterized model of a wellbore in laminated formation,said parameterized model including a plurality of laminated formationand wellbore related parameters; b) means for considering measurementdata to determine said laminated formation and wellbore relatedparameters; c) means for updating said parameterized model by adoptingsaid determined laminated formation and wellbore related parameters; andd) means for applying said updated parameterized model to derive asolution of wellbore stability.
 15. The apparatus of claim 14 furthercomprising means for repeating said considering and updating withadditional measurement data to produce a further updated parameterizedmodel to derive a further solution of wellbore stability.
 16. An articleof manufacture, comprising: a computer usable medium having a computerreadable program code means embodied therein for predicting wellborestability, the computer readable program code means in said article ofmanufacture comprising: a) computer-readable program means for creatinga parameterized model of a wellbore in laminated formation, saidparameterized model including a plurality of laminated formation andwellbore related parameters; b) computer-readable program means forconsidering measurement data to determine said laminated formation andwellbore related parameters; c) computer-readable program means forupdating said parameterized model by adopting said determined laminatedformation and wellbore related parameters; and d) computer-readableprogram means for applying said updated parameterized model to derive asolution of wellbore stability.
 17. The method of claim 16 furthercomprising computer-readable program means for repeating saidconsidering and updating with additional measurement data to produce afurther updated parameterized model to derive a further solution ofwellbore stability.